Euler's buckling load is the force at which an ideal straight column bows sideways. We set it against Hong Kong's published steel column curve b, for steel with a design strength (py) of 275 N/mm², and Euler's stress came out about 2.5 times the Code's design strength pc at a slenderness of 60, 1.4 times at 100 and 1.14 times at the Code's general limit of 200. Those ratios are ours, computed from the Code's Appendix 8.4 formulae before rounding.
They compare two published curves and they are not capacities. Nothing here says any column will carry a load, and member design belongs to a qualified engineer under the code that applies. The curve stays under Euler by construction, because Euler's stress is one of its inputs. What we measured is by how much.
Taller columns bear a quarter of the load at double the height
The result sits in the appendix on elastic curves of Euler's 1744 Methodus inveniendi (catalogue entry E65). Section 37 takes a column of height a and gives the force needed to start bending it as π²Ekk/a². That is the number a pin-ended column gives today, on our reading. The passage has no I and no K. If elasticity and thickness stay the same, it continues, in our translation, "a column twice as tall will be able to bear only a quarter of the load".
Ekk is one symbol for stiffness. Euler says an experiment should supply it and gives no measured value in sections 38 to 40. Section 40 then pulls it apart: it depends first on the material, runs in proportion to width and seems, he says, to go as the square of the thickness. A rectangle's second moment of area today carries the cube.
Slimmer columns bring Euler close to Hong Kong's curves without reaching them
A column curve plots design strength against slenderness, which is the effective length KL divided by the radius of gyration r. Hong Kong's Code of Practice for the Structural Use of Steel 2011 (2023 Edition) has five, a0 to d, assigned by section type, and Appendix 8.4 builds each from (pE − pc)(py − pc) = ηpEpc, with pE as Euler's stress. We chose this Code because its text is free to read, and did not read the codes in force elsewhere, Singapore's and Malaysia's included.
Curve b, py 275, E 205,000 N/mm², computed from Appendix 8.4, with Euler ÷ pc taken before rounding:
| Slenderness KL/r | Euler stress (N/mm²) | Curve pc (N/mm²) | Euler ÷ pc |
|---|---|---|---|
| 40 | 1,265 | 250 | 5.06 |
| 60 | 562 | 221 | 2.55 |
| 100 | 202 | 141 | 1.44 |
| 200 | 51 | 44 | 1.14 |
Our coded version matched 22 printed cells of Table 8.8(b) to within 0.49 N/mm². The ratios use values before rounding, so dividing the printed cells moves the second decimal: 1.14 becomes 1.15 at 200. Across the 13 pairings the Code offers for py 275, 355 and 460, the ratio is lowest at 200, where it runs from 1.04 (py 460, curve a0) to 1.31 (py 275, curve d). On curve b at py 275 it is still 1.03 at a slenderness of 1,000.
To run one case yourself, open our Euler buckling load calculator, keep steel py 275 (E 200 GPa), and enter I = 1,000 cm⁴, L = 3 m, pinned ends and r = 30 mm. It returns 2,193.2 kN at slenderness 100 and 197 MPa, 0.72 of the 275 strength, and stops at Euler. Our Appendix 8.4 run at E = 200 gives 138.9 N/mm², or 1,543 kN on the implied area of 11,111 mm², a ratio of 1.42. The table's 1.44 uses the Code's E.
Before its 1 November 2026 revision the calculator printed "K = [object Object]" and judged every material against a fixed 250 MPa. At E 200 that note went quiet above slenderness 88.9, with curve b at py 275 still 1.55 times under Euler. It now prints K and compares Euler's stress only with the strength on its form.
Stronger steel gains less as columns get slimmer
In the Code's Table 3.2, py 275 and py 460 are the values for S275 and S460 up to 16 mm thick, a 1.67-fold step in design strength. On curve b, before rounding, the py 460 curve gives 1.62 times the py 275 figure at slenderness 40, 1.31 times at 80, 1.19 times at 100 and 1.06 times at 200. The common wisdom that buckling is independent of material strength only really fits at the slender end; it breaks down for columns with a slenderness of 40 to 80. Euler's stress holds no strength term, so the two curves close up as both approach it.
Firmer ends add less load on Hong Kong's recommended K values
K scales the length that enters the formula. The Code's Table 8.6 prints theoretical and recommended values, and clause 8.7.2(d) says they differ "because of uncertainty in boundary assumptions". The last column shows our own arithmetic based on this single code's table.
| End conditions | Theoretical K | Recommended K | Euler load at recommended ÷ Euler load at theoretical |
|---|---|---|---|
| Fixed, fixed | 0.5 | 0.70 | 0.51 |
| Fixed, pinned | 0.7 | 0.85 | 0.68 |
| Fixed base, top rotation fixed, free to sway | 1.0 | 1.20 | 0.69 |
| Pinned, pinned | 1.0 | 1.00 | 1.00 |
| Fixed base, free top | 2.0 | 2.10 | 0.91 |
Fixing both ends of a pinned column quadruples Euler's load at K 0.5 and gives 2.04 times at K 0.70. When applied to the Code's design curve, the gain is smaller, because K is an input to slenderness. On curve b at py 275, at pinned L/r 100, we get 1.68 times at K 0.5 and 1.43 times at K 0.70.
Shorter columns leave Euler for the Johnson parabola
At low slenderness Euler's stress passes the material's strength. A parabola starting at the yield stress fy at zero slenderness and tangent to Euler touches it at half fy, at a slenderness of π√(2E/fy). That derivation is ours. This explains the two transition figures in circulation. Dropping the 2 from the slenderness formula gives the point where Euler's stress equals fy. At E 200 GPa and fy 275 MPa those are 119.8 and 84.7. At the second, Euler is 4/3 of the parabola, for any material.
Some curves do join Euler. On a figure in NASA's Astronautic Structures Manual a tangent-modulus curve runs into it, and in that range the column "buckles in the classical Euler manner". Hong Kong's Code uses no Johnson parabola.
Thinner timber posts behave like Euler's columns on a 1935 rule
The USDA's 1935 Wood Handbook divides solid rectangular wood columns by length over least dimension. Up to 11, "failure is by crushing". Beyond a boundary the handbook calls K, columns are described as "behaving essentially as Euler columns", and failure in between is mixed. For its example wood it prints 22.4, a slenderness of 77.6 by our conversion.
That is a dated rule, and we read no current timber table. It does return to section 37, where Euler wrote that his result applies chiefly to wooden columns (circa columnas ligneas), because they are prone to bending. Nearly two centuries later, a government handbook still gave him the slender end of a wooden post.
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